Multilinear Harmonic Analysis and Applications

谐波分析涉及将信号或函数分解成基波，逐个分析各个部分，然后将这些局部信息合成成关于原始对象的信息。这种方法在物理学和工程学中有广泛的应用，但在数学本身也有应用。调和分析的一个特别重点是获得有关对象的良好定量信息，这在遍历理论和组合学领域中通常可以用来产生强有力的定量结果。遍历理论研究长期演化的动力系统的行为，而组合学的一个焦点是研究大而任意集合中结构的存在。这些结构可以是几何图案，例如算术级数、正方形的顶点、它们的平移、旋转或膨胀。该项目将专注于研究调和分析中自然发生的积分变换，并建立一套工具，进一步将调和分析的范围扩展到附近的遍历理论和组合学领域。由于这些都是非常基础性的课题，来自这些领域的结果和技术适用于更广泛的科学背景。例如，在随机分析、偏微分方程、信号处理和医学成像中的应用。该项目的基础数学研究将解决社区目前相关的科学问题，继续追求理解开放的数学问题。该项目还将包括与博士后合作，指导本科生，组织研讨会和讲习班，促进交叉协作和思想交流。本项目的第一部分将讨论具有超图结构的多线性奇异积分的基本族。主要目的将是解决这些形式在勒贝格空间上的有界性。这些问题将通过扩展之前由PI和合作者开发的技术来解决。这项工作的部分灵感来自于调和分析中的一个重大开放问题，即单纯形希尔伯特变换的有界性。此外，这个项目将讨论多线性振荡积分形式及其与多线性奇异积分和具有曲率的极大函数的变量之间的联系。然后将多线性奇异积分和振荡积分的估计应用于遍历理论。在此背景下，主要目标将是研究各种类型的遍历平均沿交换保测变换的轨道的收敛，方法是通过更强的定量变分估计建立定性的收敛声明。后者将被转化为欧几里得空间中的问题，并通过在整个项目中开发的调和分析技术来解决。这些方法也将被应用于欧几里德拉姆齐理论中的问题，其中将调查欧几里德空间的大子集中是否存在线性和非线性的点配置。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Harmonic analysis is concerned with decomposing signals or functions into elementary waves, analyzing the pieces individually, and then synthesizing this local information into information about the original object. This approach has wide applications in physics and engineering, but also within mathematics itself. A special focus of harmonic analysis is to obtain good quantitative information on the objects in question, which can often be applied to yield strong quantitative results in the fields of ergodic theory and combinatorics. Ergodic theory investigates behavior of dynamical systems that evolve for a long time, while one focus of combinatorics is to study existence of structures in large but otherwise arbitrary sets. These structures can be geometric patterns such as arithmetic progressions, vertices of a square, their translates, rotates, or dilates. The project will focus on the study of integral transformations that naturally occur within harmonic analysis and build a set of tools that further expands the reach of harmonic analysis into the nearby fields of ergodic theory and combinatorics. As these are very foundational subjects, the results and techniques from these areas are applicable in a wider scientific context. Examples include applications in stochastic analysis, partial differential equations, signal processing, and medical imaging. The project’s fundamental mathematical research will address the community’s currently relevant scientific questions in its continued pursuit in understanding open mathematical questions. The project will further entail working with a postdoc and mentoring undergraduate students, organization of seminars and workshops that promote cross-collaboration and exchange of ideas. The first part of this project will deal with a fundamental family of multilinear singular integral forms with a hypergraph structure. The main goal will be to address boundedness of these forms on Lebesgue spaces. These questions will be approached by extending the techniques previously developed by the PI and collaborators. The work is partially inspired by a major open question in harmonic analysis, boundedness of the simplex Hilbert transforms. Further, this project will address multilinear oscillatory integral forms and their connections with variants of multilinear singular integrals and maximal functions with curvature. Estimates on multilinear singular and oscillatory integrals will then be applied to ergodic theory. In this context, the main goal will be to investigate convergence of various types of ergodic averages along the orbits of commuting measure-preserving transformations by establishing qualitative convergence statements via stronger quantitative variational estimates. The latter will be translated to questions in Euclidean spaces and addressed with harmonic analysis techniques developed throughout this project. These methods will also be applied to questions in Euclidean Ramsey theory, where existence of linear and non-linear point configurations in large subsets of the Euclidean space will be investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.